What is the universal covering of surface with genus 2

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I'm looking for the universal covering for any closed surface, and I gusse that we need only to find the universal covering of $S^2$, $M_1$, and $M_2$, where $M_j$ is the closed surface of genus $j$. The only one I need to consider is the covering of $M_2$.

Answer 1

While the existence of a universal covering space is a reasonably elementary theorem (if rather abstract and inscrutable), specific constructions of universal covering spaces are often not at all obvious.

In this case, the idea --- that may not be at all obvious at first --- is to use hyperbolic geometry, which you learn about when studying either differential geometry, or geometric topology, or complex analysis. Here's the chain of ideas:

• $M_2$ has a complete hyperbolic metric, giving it the structure of a hyperbolic surface.
• For any hyperbolic surface $S$, there exists a universal covering map $\mathbb H^2 \to S$, where $\mathbb H^2$ is the hyperbolic plane.
• $\mathbb H^2$ is homeomorphic to any of the following spaces: the open upper half plane $\{(x,y) \in \mathbb R^2 \mid y > 0\}$; the open unit disc in $\mathbb R^2$; $\mathbb R^2$ itself.

As a consequence, one may use any of those spaces as a universal covering space of $M_2$.

It is reasonably elementary to construct an explicit hyperbolic structure on $M_2$, although I can't believe I have to link to physics.stackexchange to get a good picture!! But in fact any connected surface has a hyperbolic structure unless it is homeomorphic to the sphere, the projective plane, the torus, or the Klein bottle (this is an application of the uniformization theorem, which is where complex analysis comes in). The torus and the Klein bottle each have Euclidean structures and so their universal covering spaces are also homeomorphic to $\mathbb R^2$.

There are several good books which introduce you to hyperbolic geometry and its applications to geometric topology.